Question

The Organization for the Promotion of Brussels Sprouts has convinced the government of Ironia to institute a price floor on the sale of brussels sprouts, at \(\$ 8\) per bushel. Demand is given by \(\mathrm{P}=9-\mathrm{Q}\) and supply by \(\mathrm{P}=2 \mathrm{Q}\), where \(\mathrm{Q}\) is measured in thousands of bushels. [LO 6.2\(]\) a. What will be the price and quantity of brussels sprouts sold at market equilibrium? b. What will be the price and quantity sold with the price floor? c. How big will be the excess supply of brussels sprouts produced with the price floor?

Short answer

a) Price: 6, Quantity: 3k bushels; b) Price: 8, Quantity demanded: 1k, supplied: 4k; c) Excess supply: 3k bushels.

Step-by-step solution

Find Equilibrium Quantity and Price

To determine the market equilibrium, equate the demand and supply equations: \[ 9 - Q = 2Q \] Solve for \( Q \):\[ 9 = 3Q \] \[ Q = 3 \] Now, substitute \( Q = 3 \) into either the demand or the supply equation to find \( P \):Using the demand equation:\[ P = 9 - 3 = 6 \] Thus, the equilibrium price is \( 6 \) and the equilibrium quantity is \( 3 \) thousand bushels.

Effect of Price Floor on Quantity Demanded and Supplied

A price floor of \$8 means the price is set above the equilibrium price. Substitute \( P = 8 \) into the demand and supply equations to find the new quantities:For demand:\[ 8 = 9 - Q_d \] \[ Q_d = 1 \]For supply:\[ 8 = 2Q_s \] \[ Q_s = 4 \] The price floor results in a quantity demanded of \( 1 \) thousand bushels and a quantity supplied of \( 4 \) thousand bushels.

Calculate Excess Supply

Excess supply is the difference between the quantity supplied and the quantity demanded. Using our previous results:\[ \text{Excess Supply} = Q_s - Q_d = 4 - 1 = 3 \] There will be an excess supply of \( 3 \) thousand bushels of brussels sprouts due to the price floor.

Key concepts

Market Equilibrium

Market equilibrium occurs in a situation where the quantity of a product demanded by consumers is equal to the quantity of the product supplied by producers. At this point, the market is "at rest," and there's no tendency for the price to change, as neither excess supply nor excess demand exists.

In the Brussels sprouts example, the demand and supply equations are given as \( \text{P} = 9 - \text{Q} \) and \( \text{P} = 2\text{Q} \) respectively. To find the market equilibrium, we equate these equations:
\[ 9 - Q = 2Q \]
Solve for \( Q \) to discover that the equilibrium quantity is \( Q = 3 \) thousand bushels. Substituting this value into the demand equation, we find the equilibrium price \( P \) is \( 6 \).

• Equilibrium Price = $6 per bushel
• Equilibrium Quantity = 3 thousand bushels

At this equilibrium, the price and quantity of Brussels sprouts allow market forces to balance perfectly.

Excess Supply

Excess supply, also known as a surplus, happens when the quantity supplied of a good exceeds the quantity demanded at a given price level. This typically occurs when a price floor is introduced, holding the price above the market equilibrium.

With a price floor of \(8, higher than the equilibrium price of \)6, producers are willing to supply more Brussels sprouts than consumers are willing to buy. Let's look at the calculations:

• Quantity Demanded at \(8 (Q_d): From the demand equation, \( 8 = 9 - Q_d \) gives \( Q_d = 1 \) thousand bushels.
• Quantity Supplied at \)8 (Q_s): From the supply equation, \( 8 = 2Q_s \) results in \( Q_s = 4 \) thousand bushels.
• Excess Supply: \( Q_s - Q_d = 4 - 1 = 3 \) thousand bushels.

As a result, there is an excess supply of 3 thousand bushels, leading to unsold Brussels sprouts piling up.

Demand and Supply Equations

Demand and supply equations are mathematical representations of the relationship between the price of a commodity and the quantity demanded or supplied. These equations help us understand and predict how different factors affect market behavior.

The demand equation, \( \text{P} = 9 - \text{Q} \), suggests that as the price \( P \) of Brussels sprouts decreases, the quantity demanded \( Q \) increases. This inverse relationship is typical in most markets, as consumers tend to buy more when prices are lower.

• Equation Form: P = 9 - Q
• Inverse Relationship: As P decreases, Q increases

On the other side, the supply equation \( \text{P} = 2\text{Q} \) indicates a direct relationship between price and supply quantity. As prices increase, producers are willing to supply more bushels of Brussels sprouts.

• Equation Form: P = 2Q
• Direct Relationship: As P increases, Q_s increases

By analyzing these equations, we grasp how pricing strategies like price floors can disturb natural market equilibrium, causing situations like excess supply.